MATH 121 - ALGEBRA - Lecture 1 Notes

MATH 121 - ALGEBRA

  • Samuel N. N. Norte

Lecture 1: Algebraic Expressions & Operations

Definition: What is Algebra?

  • Algebra is often described as "arithmetic with letters."
  • It differs from arithmetic as it deals with variables that extend the generality and scope of arithmetic.

Algebraic Quantities

  • The most distinguishing feature of algebra is the use of letters to represent numerical quantities.
  • Simple symbols represent mathematical operations.
  • Origin: System based on René Descartes' methods (17th century).
  • Letters, particularly x, y, and z, signify unknown quantities (variables).

Mathematical Operations

  • Signs of operations:
    • + (plus) indicates addition.
    • - (minus) indicates subtraction.
    • Adjacent letters signify multiplication.
      • Example: (ax)(ax) signifies the product of a and x.

Algebraic Expression

  • Definition: An algebraic expression comprises a group of numbers, symbols, parentheses, and variables that express an operation or a series of operations.
  • Examples of algebraic expressions:
    • (m+8)(m + 8)
    • (a)(b)(a)(b)
    • (3m+6n−6)(3m + 6n - 6)
    • (r−3)(r - 3)

Coefficients

  • Definition: A coefficient is the number multiplied by the variable in an algebraic expression.
  • Examples:
    • In (6m+5)(6m + 5), the coefficient is 6.
    • In (8r+7m+4)(8r + 7m + 4), the coefficients are 8 and 7.
    • In (14b−8)(14b - 8), the coefficient is 14.

Terms

  • Definition: A term is the name given to a number, a variable, or a combination of a number and a variable by multiplication or division.
  • Examples:
    • Expression (a+2)(a + 2) consists of terms: a, 2.
    • Expression (3m+6n−6)(3m + 6n - 6) consists of terms: (3m,6n,−6)(3m, 6n, -6)

Constants

  • Definition: A constant is a number whose value cannot change.
  • Example: In the expression (5x+7y−2)(5x + 7y - 2), the constant is -2.

Identify Terms, Coefficients, and Constants

  • Exercises: Identify the terms, coefficients, and constants in the following expressions:
    1. (12a−6b+4)(12a - 6b + 4)
    2. (4x−2y)(4x - 2y)
    3. (c−32)(c - 32)
    4. (3x+2)(3x + 2)

Writing Algebraic Expressions

  • You can translate word phrases into variables.
  • Examples:
    1. "Three more than a number" = (x+3)(x + 3)
    2. "The quotient of a number and 8" = (y/8)(y/8)
    3. "Six times a number" = (6n)(6n)
    4. "Fifteen less than a number" = (z−15)(z - 15)
    5. "The quotient of 30 and a number plus 10" = (30/x+10)(30/x + 10)

Evaluating Algebraic Expressions

  • Definition: Evaluating involves finding the value of an algebraic expression by substituting numbers for variables.
  • Example 1:
    • Expression: (m+8)(m + 8)
    • For (m=2)(m = 2): (2+8=10)(2 + 8 = 10)
  • Example 2:
    • Expression: (r−3)(r - 3)
    • For (r=5)(r = 5): (5−3=2)(5 - 3 = 2)

Simplifying Algebraic Expressions

  • Definition: Simplifying means combining like terms and completing all operations.
  • Example:
    • Expression: (m+8+m)(m + 8 + m) with (m=2)(m = 2) results in
      • Combine: (2m+8=2(2)+8=4+8=12)(2m + 8 = 2(2) + 8 = 4 + 8 = 12)

Words Leading to Addition & Subtraction

  • Addition Keywords:
    • Add
    • Plus
    • Sum
    • Total
    • Increased by
    • More than
  • Subtraction Keywords:
    • Minus
    • Difference
    • Subtract
    • Less than
    • Decreased by

Multiplication and Division Words

  • Multiplication Keywords:
    • Product
    • Times
    • Multiply
  • Division Keywords:
    • Quotient
    • Divide

Writing Algebraic Expressions for Different Phrases

  • Examples for word phrases:
    1. "Ten more than a number" = (n+10)(n + 10)
    2. "A number decreased by 5" = (w−5)(w - 5)
    3. "Six less than a number" = (x−6)(x - 6)
    4. "A number increased by 8" = (n+8)(n + 8)
    5. "The sum of a number & 9" = (n+9)(n + 9)
    6. Similar examples for more phrases.

Problem Sets for Algebraic Expressions

  • Example problems to write algebraic expressions:
    1. "A quarter more than r" = (r+0.25)(r + 0.25)
    2. "The product of 2 and q" = (2q)(2q)
    3. "Nine less than y" = (y−9)(y - 9)
    4. Expressions involve divisions and combinations of numbers and variables.

Absolute Values

  • Definition: For any real number x, the absolute value of x (denoted as |x|) is defined as follows:
    • If (x≥0(x \geq 0) then (∣x∣=x)(|x| = x)
    • If (x < 0) then (∣x∣=−x)(|x| = -x)
  • Examples:
    • (∣5∣=5,∣0∣=0,∣−5∣=5)(|5| = 5, |0| = 0, |-5| = 5)

Generalization on Absolute Values

  • If ∣x∣=a|x| = a, where a > 0, then (x=a)(x = a) or (x=−a)(x = -a)

Summary

  • The key concepts covered include:
    • Definition of Algebra
    • Variables
    • Algebraic Expressions
    • Coefficients
    • Terms
    • Constants
    • Simplifying and Evaluating Algebraic Expressions
    • Absolute Value Functions

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